/
anneal.py
475 lines (406 loc) · 15.8 KB
/
anneal.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
# Original Author: Travis Oliphant 2002
# Bug-fixes in 2006 by Tim Leslie
from __future__ import division, print_function, absolute_import
import numpy
from numpy import asarray, tan, exp, ones, squeeze, sign, \
all, log, sqrt, pi, shape, array, minimum, where, random
from .optimize import Result, _check_unknown_options
from scipy.lib.six.moves import xrange
__all__ = ['anneal']
_double_min = numpy.finfo(float).min
_double_max = numpy.finfo(float).max
class base_schedule(object):
def __init__(self):
self.dwell = 20
self.learn_rate = 0.5
self.lower = -10
self.upper = 10
self.Ninit = 50
self.accepted = 0
self.tests = 0
self.feval = 0
self.k = 0
self.T = None
def init(self, **options):
self.__dict__.update(options)
self.lower = asarray(self.lower)
self.lower = where(self.lower == numpy.NINF, -_double_max, self.lower)
self.upper = asarray(self.upper)
self.upper = where(self.upper == numpy.PINF, _double_max, self.upper)
self.k = 0
self.accepted = 0
self.feval = 0
self.tests = 0
def getstart_temp(self, best_state):
""" Find a matching starting temperature and starting parameters vector
i.e. find x0 such that func(x0) = T0.
Parameters
----------
best_state : _state
A _state object to store the function value and x0 found.
Returns
-------
x0 : array
The starting parameters vector.
"""
assert(not self.dims is None)
lrange = self.lower
urange = self.upper
fmax = _double_min
fmin = _double_max
for _ in range(self.Ninit):
x0 = random.uniform(size=self.dims)*(urange-lrange) + lrange
fval = self.func(x0, *self.args)
self.feval += 1
if fval > fmax:
fmax = fval
if fval < fmin:
fmin = fval
best_state.cost = fval
best_state.x = array(x0)
self.T0 = (fmax-fmin)*1.5
return best_state.x
def accept_test(self, dE):
T = self.T
self.tests += 1
if dE < 0:
self.accepted += 1
return 1
p = exp(-dE*1.0/self.boltzmann/T)
if (p > random.uniform(0.0, 1.0)):
self.accepted += 1
return 1
return 0
def update_guess(self, x0):
pass
def update_temp(self, x0):
pass
# A schedule due to Lester Ingber
class fast_sa(base_schedule):
def init(self, **options):
self.__dict__.update(options)
if self.m is None:
self.m = 1.0
if self.n is None:
self.n = 1.0
self.c = self.m * exp(-self.n * self.quench)
def update_guess(self, x0):
x0 = asarray(x0)
u = squeeze(random.uniform(0.0, 1.0, size=self.dims))
T = self.T
y = sign(u-0.5)*T*((1+1.0/T)**abs(2*u-1)-1.0)
xc = y*(self.upper - self.lower)
xnew = x0 + xc
return xnew
def update_temp(self):
self.T = self.T0*exp(-self.c * self.k**(self.quench))
self.k += 1
return
class cauchy_sa(base_schedule):
def update_guess(self, x0):
x0 = asarray(x0)
numbers = squeeze(random.uniform(-pi/2, pi/2, size=self.dims))
xc = self.learn_rate * self.T * tan(numbers)
xnew = x0 + xc
return xnew
def update_temp(self):
self.T = self.T0/(1+self.k)
self.k += 1
return
class boltzmann_sa(base_schedule):
def update_guess(self, x0):
std = minimum(sqrt(self.T)*ones(self.dims), (self.upper-self.lower)/3.0/self.learn_rate)
x0 = asarray(x0)
xc = squeeze(random.normal(0, 1.0, size=self.dims))
xnew = x0 + xc*std*self.learn_rate
return xnew
def update_temp(self):
self.k += 1
self.T = self.T0 / log(self.k+1.0)
return
class _state(object):
def __init__(self):
self.x = None
self.cost = None
# TODO:
# allow for general annealing temperature profile
# in that case use update given by alpha and omega and
# variation of all previous updates and temperature?
# Simulated annealing
def anneal(func, x0, args=(), schedule='fast', full_output=0,
T0=None, Tf=1e-12, maxeval=None, maxaccept=None, maxiter=400,
boltzmann=1.0, learn_rate=0.5, feps=1e-6, quench=1.0, m=1.0, n=1.0,
lower=-100, upper=100, dwell=50, disp=True):
"""
Minimize a function using simulated annealing.
Schedule is a schedule class implementing the annealing schedule.
Available ones are 'fast', 'cauchy', 'boltzmann'
Parameters
----------
func : callable ``f(x, *args)``
Function to be optimized.
x0 : ndarray
Initial guess.
args : tuple
Extra parameters to `func`.
schedule : base_schedule
Annealing schedule to use (a class).
full_output : bool
Whether to return optional outputs.
T0 : float
Initial Temperature (estimated as 1.2 times the largest
cost-function deviation over random points in the range).
Tf : float
Final goal temperature.
maxeval : int
Maximum function evaluations.
maxaccept : int
Maximum changes to accept.
maxiter : int
Maximum cooling iterations.
learn_rate : float
Scale constant for adjusting guesses.
boltzmann : float
Boltzmann constant in acceptance test
(increase for less stringent test at each temperature).
feps : float
Stopping relative error tolerance for the function value in
last four coolings.
quench, m, n : float
Parameters to alter fast_sa schedule.
lower, upper : float or ndarray
Lower and upper bounds on `x`.
dwell : int
The number of times to search the space at each temperature.
disp : bool
Set to True to print convergence messages.
Returns
-------
xmin : ndarray
Point giving smallest value found.
Jmin : float
Minimum value of function found.
T : float
Final temperature.
feval : int
Number of function evaluations.
iters : int
Number of cooling iterations.
accept : int
Number of tests accepted.
retval : int
Flag indicating stopping condition:
0 : Points no longer changing
1 : Cooled to final temperature
2 : Maximum function evaluations
3 : Maximum cooling iterations reached
4 : Maximum accepted query locations reached
5 : Final point not the minimum amongst encountered points
See also
--------
minimize: Interface to minimization algorithms for multivariate
functions. See the 'Anneal' `method` in particular.
Notes
-----
Simulated annealing is a random algorithm which uses no derivative
information from the function being optimized. In practice it has
been more useful in discrete optimization than continuous
optimization, as there are usually better algorithms for continuous
optimization problems.
Some experimentation by trying the difference temperature
schedules and altering their parameters is likely required to
obtain good performance.
The randomness in the algorithm comes from random sampling in numpy.
To obtain the same results you can call numpy.random.seed with the
same seed immediately before calling scipy.optimize.anneal.
We give a brief description of how the three temperature schedules
generate new points and vary their temperature. Temperatures are
only updated with iterations in the outer loop. The inner loop is
over loop over xrange(dwell), and new points are generated for
every iteration in the inner loop. (Though whether the proposed
new points are accepted is probabilistic.)
For readability, let d denote the dimension of the inputs to func.
Also, let x_old denote the previous state, and k denote the
iteration number of the outer loop. All other variables not
defined below are input variables to scipy.optimize.anneal itself.
In the 'fast' schedule the updates are ::
u ~ Uniform(0, 1, size=d)
y = sgn(u - 0.5) * T * ((1+ 1/T)**abs(2u-1) -1.0)
xc = y * (upper - lower)
x_new = x_old + xc
c = n * exp(-n * quench)
T_new = T0 * exp(-c * k**quench)
In the 'cauchy' schedule the updates are ::
u ~ Uniform(-pi/2, pi/2, size=d)
xc = learn_rate * T * tan(u)
x_new = x_old + xc
T_new = T0 / (1+k)
In the 'boltzmann' schedule the updates are ::
std = minimum( sqrt(T) * ones(d), (upper-lower) / (3*learn_rate) )
y ~ Normal(0, std, size=d)
x_new = x_old + learn_rate * y
T_new = T0 / log(1+k)
"""
opts = {'schedule' : schedule,
'T0' : T0,
'Tf' : Tf,
'maxfev' : maxeval,
'maxaccept' : maxaccept,
'maxiter' : maxiter,
'boltzmann' : boltzmann,
'learn_rate': learn_rate,
'ftol' : feps,
'quench' : quench,
'm' : m,
'n' : n,
'lower' : lower,
'upper' : upper,
'dwell' : dwell,
'disp' : disp}
res = _minimize_anneal(func, x0, args, **opts)
if full_output:
return res['x'], res['fun'], res['T'], res['nfev'], res['nit'], \
res['accept'], res['status']
else:
return res['x'], res['status']
def _minimize_anneal(func, x0, args=(),
schedule='fast', T0=None, Tf=1e-12, maxfev=None,
maxaccept=None, maxiter=400, boltzmann=1.0, learn_rate=0.5,
ftol=1e-6, quench=1.0, m=1.0, n=1.0, lower=-100,
upper=100, dwell=50, disp=False,
**unknown_options):
"""
Minimization of scalar function of one or more variables using the
simulated annealing algorithm.
Options for the simulated annealing algorithm are:
disp : bool
Set to True to print convergence messages.
schedule : str
Annealing schedule to use. One of: 'fast', 'cauchy' or
'boltzmann'.
T0 : float
Initial Temperature (estimated as 1.2 times the largest
cost-function deviation over random points in the range).
Tf : float
Final goal temperature.
maxfev : int
Maximum number of function evaluations to make.
maxaccept : int
Maximum changes to accept.
maxiter : int
Maximum number of iterations to perform.
boltzmann : float
Boltzmann constant in acceptance test (increase for less
stringent test at each temperature).
learn_rate : float
Scale constant for adjusting guesses.
ftol : float
Relative error in ``fun(x)`` acceptable for convergence.
quench, m, n : float
Parameters to alter fast_sa schedule.
lower, upper : float or ndarray
Lower and upper bounds on `x`.
dwell : int
The number of times to search the space at each temperature.
This function is called by the `minimize` function with
`method=anneal`. It is not supposed to be called directly.
"""
_check_unknown_options(unknown_options)
maxeval = maxfev
feps = ftol
x0 = asarray(x0)
lower = asarray(lower)
upper = asarray(upper)
schedule = eval(schedule+'_sa()')
# initialize the schedule
schedule.init(dims=shape(x0),func=func,args=args,boltzmann=boltzmann,T0=T0,
learn_rate=learn_rate, lower=lower, upper=upper,
m=m, n=n, quench=quench, dwell=dwell)
current_state, last_state, best_state = _state(), _state(), _state()
if T0 is None:
x0 = schedule.getstart_temp(best_state)
else:
best_state.x = None
best_state.cost = numpy.Inf
last_state.x = asarray(x0).copy()
fval = func(x0,*args)
schedule.feval += 1
last_state.cost = fval
if last_state.cost < best_state.cost:
best_state.cost = fval
best_state.x = asarray(x0).copy()
schedule.T = schedule.T0
fqueue = [100, 300, 500, 700]
iters = 0
while 1:
for n in xrange(dwell):
current_state.x = schedule.update_guess(last_state.x)
current_state.cost = func(current_state.x,*args)
schedule.feval += 1
dE = current_state.cost - last_state.cost
if schedule.accept_test(dE):
last_state.x = current_state.x.copy()
last_state.cost = current_state.cost
if last_state.cost < best_state.cost:
best_state.x = last_state.x.copy()
best_state.cost = last_state.cost
schedule.update_temp()
iters += 1
# Stopping conditions
# 0) last saved values of f from each cooling step
# are all very similar (effectively cooled)
# 1) Tf is set and we are below it
# 2) maxeval is set and we are past it
# 3) maxiter is set and we are past it
# 4) maxaccept is set and we are past it
fqueue.append(squeeze(last_state.cost))
fqueue.pop(0)
af = asarray(fqueue)*1.0
if all(abs((af-af[0])/af[0]) < feps):
retval = 0
if abs(af[-1]-best_state.cost) > feps*10:
retval = 5
if disp:
print("Warning: Cooled to %f at %s but this is not" \
% (squeeze(last_state.cost),
str(squeeze(last_state.x))) \
+ " the smallest point found.")
break
if (Tf is not None) and (schedule.T < Tf):
retval = 1
break
if (maxeval is not None) and (schedule.feval > maxeval):
retval = 2
break
if (iters > maxiter):
if disp:
print("Warning: Maximum number of iterations exceeded.")
retval = 3
break
if (maxaccept is not None) and (schedule.accepted > maxaccept):
retval = 4
break
result = Result(x=best_state.x, fun=best_state.cost,
T=schedule.T, nfev=schedule.feval, nit=iters,
accept=schedule.accepted, status=retval,
success=(retval <= 1),
message={0: 'Points no longer changing',
1: 'Cooled to final temperature',
2: 'Maximum function evaluations',
3: 'Maximum cooling iterations reached',
4: 'Maximum accepted query locations reached',
5: 'Final point not the minimum amongst '
'encountered points'}[retval])
return result
if __name__ == "__main__":
from numpy import cos
# minimum expected at ~-0.195
func = lambda x: cos(14.5*x-0.3) + (x+0.2)*x
print(anneal(func,1.0,full_output=1,upper=3.0,lower=-3.0,feps=1e-4,maxiter=2000,schedule='cauchy'))
print(anneal(func,1.0,full_output=1,upper=3.0,lower=-3.0,feps=1e-4,maxiter=2000,schedule='fast'))
print(anneal(func,1.0,full_output=1,upper=3.0,lower=-3.0,feps=1e-4,maxiter=2000,schedule='boltzmann'))
# minimum expected at ~[-0.195, -0.1]
func = lambda x: cos(14.5*x[0]-0.3) + (x[1]+0.2)*x[1] + (x[0]+0.2)*x[0]
print(anneal(func,[1.0, 1.0],full_output=1,upper=[3.0, 3.0],lower=[-3.0, -3.0],feps=1e-4,maxiter=2000,schedule='cauchy'))
print(anneal(func,[1.0, 1.0],full_output=1,upper=[3.0, 3.0],lower=[-3.0, -3.0],feps=1e-4,maxiter=2000,schedule='fast'))
print(anneal(func,[1.0, 1.0],full_output=1,upper=[3.0, 3.0],lower=[-3.0, -3.0],feps=1e-4,maxiter=2000,schedule='boltzmann'))